What Is The Sum?${ \frac{3y}{y^2+7y+10} + \frac{2}{y+2} + \frac{5}{y-5} + \frac{5(y+2)}{(y-2)(y+5)} + \frac{5}{y+5} + \frac{5(y-2)}{(y-5)(y+2)} }$

by ADMIN 148 views

Introduction

When dealing with complex fractions, it can be challenging to simplify them and find their sum. Complex fractions involve multiple fractions within a single expression, making it difficult to evaluate their value. In this article, we will explore a step-by-step guide on how to simplify complex fractions and evaluate the sum of the given expression.

Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions within its numerator or denominator. It can be represented as:

f(x)g(x){ \frac{f(x)}{g(x)} }

where f(x) and g(x) are fractions. Complex fractions can be simplified by finding a common denominator and combining the fractions.

Simplifying the Given Expression

The given expression is:

3yy2+7y+10+2y+2+5y−5+5(y+2)(y−2)(y+5)+5y+5+5(y−2)(y−5)(y+2){ \frac{3y}{y^2+7y+10} + \frac{2}{y+2} + \frac{5}{y-5} + \frac{5(y+2)}{(y-2)(y+5)} + \frac{5}{y+5} + \frac{5(y-2)}{(y-5)(y+2)} }

To simplify this expression, we need to find a common denominator for all the fractions.

Finding a Common Denominator

The common denominator for the given expression is the product of all the denominators:

(y2+7y+10)(y+2)(y−5)(y+5){ (y^2+7y+10)(y+2)(y-5)(y+5) }

However, this is not the most efficient way to find the common denominator. We can simplify the expression by factoring the denominators and finding a common factor.

Factoring the Denominators

The denominators can be factored as follows:

y2+7y+10=(y+2)(y+5){ y^2+7y+10 = (y+2)(y+5) } y+2=y+2{ y+2 = y+2 } y−5=y−5{ y-5 = y-5 } y+5=y+5{ y+5 = y+5 } (y−2)(y+5)=(y−2)(y+5){ (y-2)(y+5) = (y-2)(y+5) } (y−5)(y+2)=(y−5)(y+2){ (y-5)(y+2) = (y-5)(y+2) }

Finding a Common Factor

Now that we have factored the denominators, we can find a common factor. The common factor is the product of all the factors:

(y+2)(y+5)(y−5)(y−2){ (y+2)(y+5)(y-5)(y-2) }

Simplifying the Expression

Now that we have found the common factor, we can simplify the expression by combining the fractions.

3y(y+2)(y+5)+2y+2+5y−5+5(y+2)(y−2)(y+5)+5y+5+5(y−2)(y−5)(y+2){ \frac{3y}{(y+2)(y+5)} + \frac{2}{y+2} + \frac{5}{y-5} + \frac{5(y+2)}{(y-2)(y+5)} + \frac{5}{y+5} + \frac{5(y-2)}{(y-5)(y+2)} }

Combining the Fractions

To combine the fractions, we need to find a common denominator for each pair of fractions.

3y(y−5)(y+2)(y+2)(y+5)(y−5)(y−2)+2(y+5)(y−5)(y+2)(y+5)(y−5)+5(y+2)(y−2)(y−5)(y+2)(y−2)+5(y+2)(y+5)(y−2)(y+5)(y+2)+5(y−2)(y+5)(y+5)(y−2)(y+5)+5(y−5)(y+2)(y−5)(y+2)(y−5){ \frac{3y(y-5)(y+2)}{(y+2)(y+5)(y-5)(y-2)} + \frac{2(y+5)(y-5)}{(y+2)(y+5)(y-5)} + \frac{5(y+2)(y-2)}{(y-5)(y+2)(y-2)} + \frac{5(y+2)(y+5)}{(y-2)(y+5)(y+2)} + \frac{5(y-2)(y+5)}{(y+5)(y-2)(y+5)} + \frac{5(y-5)(y+2)}{(y-5)(y+2)(y-5)} }

Simplifying the Expression

Now that we have combined the fractions, we can simplify the expression by canceling out any common factors.

3y(y−5)(y+2)(y+2)(y+5)(y−5)(y−2)+2(y+5)(y−5)(y+2)(y+5)(y−5)+5(y+2)(y−2)(y−5)(y+2)(y−2)+5(y+2)(y+5)(y−2)(y+5)(y+2)+5(y−2)(y+5)(y+5)(y−2)(y+5)+5(y−5)(y+2)(y−5)(y+2)(y−5){ \frac{3y(y-5)(y+2)}{(y+2)(y+5)(y-5)(y-2)} + \frac{2(y+5)(y-5)}{(y+2)(y+5)(y-5)} + \frac{5(y+2)(y-2)}{(y-5)(y+2)(y-2)} + \frac{5(y+2)(y+5)}{(y-2)(y+5)(y+2)} + \frac{5(y-2)(y+5)}{(y+5)(y-2)(y+5)} + \frac{5(y-5)(y+2)}{(y-5)(y+2)(y-5)} }

Canceling Out Common Factors

We can cancel out any common factors between the numerator and denominator.

3y(y−2)+2(y−2)+5(y−2)+5(y−2)+5(y−2)+5(y−2){ \frac{3y}{(y-2)} + \frac{2}{(y-2)} + \frac{5}{(y-2)} + \frac{5}{(y-2)} + \frac{5}{(y-2)} + \frac{5}{(y-2)} }

Simplifying the Expression

Now that we have canceled out any common factors, we can simplify the expression by combining the fractions.

3y+2+5+5+5+5(y−2){ \frac{3y+2+5+5+5+5}{(y-2)} }

Evaluating the Sum

Now that we have simplified the expression, we can evaluate the sum.

3y+18(y−2){ \frac{3y+18}{(y-2)} }

Conclusion

In this article, we have explored a step-by-step guide on how to simplify complex fractions and evaluate the sum of the given expression. We have used factoring and canceling out common factors to simplify the expression and evaluate the sum. This guide can be applied to any complex fraction, making it a valuable resource for anyone dealing with complex fractions.

Final Answer

The final answer is:

3y+18(y−2){ \frac{3y+18}{(y-2)} }

Introduction

Simplifying complex fractions can be a challenging task, but with the right guidance, it can be made easier. In this article, we will answer some of the most frequently asked questions about simplifying complex fractions.

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions within its numerator or denominator. It can be represented as:

f(x)g(x){ \frac{f(x)}{g(x)} }

where f(x) and g(x) are fractions.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to find a common denominator for all the fractions and then combine them. You can also use factoring and canceling out common factors to simplify the expression.

Q: What is the common denominator of a complex fraction?

A: The common denominator of a complex fraction is the product of all the denominators. However, this is not the most efficient way to find the common denominator. You can simplify the expression by factoring the denominators and finding a common factor.

Q: How do I factor the denominators of a complex fraction?

A: To factor the denominators of a complex fraction, you need to identify the factors of each denominator. You can use the distributive property to factor the denominators.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that a single operation can be distributed over multiple terms. It can be represented as:

a(b+c)=ab+ac{ a(b+c) = ab + ac }

Q: How do I use the distributive property to factor the denominators?

A: To use the distributive property to factor the denominators, you need to identify the factors of each denominator and then distribute the terms.

Q: What is the difference between a common factor and a common denominator?

A: A common factor is a factor that is common to both the numerator and denominator of a fraction. A common denominator is the product of all the denominators of a complex fraction.

Q: How do I cancel out common factors in a complex fraction?

A: To cancel out common factors in a complex fraction, you need to identify the common factors and then cancel them out.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is:

3y+18(y−2){ \frac{3y+18}{(y-2)} }

Q: Can I use this guide to simplify any complex fraction?

A: Yes, you can use this guide to simplify any complex fraction. The steps outlined in this guide can be applied to any complex fraction, making it a valuable resource for anyone dealing with complex fractions.

Q: What are some common mistakes to avoid when simplifying complex fractions?

A: Some common mistakes to avoid when simplifying complex fractions include:

  • Not finding a common denominator
  • Not factoring the denominators
  • Not canceling out common factors
  • Not combining the fractions

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying complex fractions. We have provided a step-by-step guide on how to simplify complex fractions and have highlighted some common mistakes to avoid. This guide can be applied to any complex fraction, making it a valuable resource for anyone dealing with complex fractions.

Final Answer

The final answer is:

3y+18(y−2){ \frac{3y+18}{(y-2)} }